is continuous and differentiable at . By composition, the case is analytic everywhere it is defined, and integrating around 0 in the complex plane yields 0 and suggests that there is no pole there. Furthermore, working from the Taylor series of , the case can be seen to have a series that is defined at 0 and equal to 1:
From this we can also see that is analytic and infinitely differentiable.
All taken together, it seems strange that such a nicely behaved function would have an irremovable "patch" necessary at . Nonetheless, I have not been able to come up with another way to write the function that allows it to be defined at zero without a special case there. Is it impossible to write this analytic function without a case block?